A Hybrid Mesh Free Local RBF- Cartesian FD Scheme for Incompressible Flow around Solid Bodies
A method for simulating flow around the solid bodies has been presented using hybrid meshfree and mesh-based schemes. The presented scheme optimizes the computational efficiency by combining the advantages of both meshfree and mesh-based methods. In this approach, a cloud of meshfree nodes has been used in the domain around the solid body. These meshfree nodes have the ability to efficiently adapt to complex geometrical shapes. In the rest of the domain, conventional Cartesian grid has been used beyond the meshfree cloud. Complex geometrical shapes can therefore be dealt efficiently by using meshfree nodal cloud and computational efficiency is maintained through the use of conventional mesh-based scheme on Cartesian grid in the larger part of the domain. Spatial discretization of meshfree nodes has been achieved through local radial basis functions in finite difference mode (RBF-FD). Conventional finite difference scheme has been used in the Cartesian ‘meshed’ domain. Accuracy tests of the hybrid scheme have been conducted to establish the order of accuracy. Numerical tests have been performed by simulating two dimensional steady and unsteady incompressible flows around cylindrical object. Steady flow cases have been run at Reynolds numbers of 10, 20 and 40 and unsteady flow problems have been studied at Reynolds numbers of 100 and 200. Flow Parameters including lift, drag, vortex shedding, and vorticity contours are calculated. Numerical results have been found to be in good agreement with computational and experimental results available in the literature.
[1] S. Dennis, and G.-Z. Chang, “Numerical solutions for steady flow past a
circular cylinder at Reynolds numbers up to 100,” J. Fluid Mech, vol.
42, no. 3, pp. 471-489, 1970.
[2] M. Braza, P. Chassaing, and H. H. Minh, “Numerical Study and
Physical Analysis of the Pressure and Velocity-Fields in the near Wake of a Circular-Cylinder,” Journal of Fluid Mechanics, vol. 165, pp. 79-
130, Apr, 1986.
[3] H. Takami, and H. B. Keller, “Steady Two‐Dimensional Viscous Flow
of an Incompressible Fluid past a Circular Cylinder,” Physics of Fluids,
vol. 12, no. 12, pp. II-51-II-56, 1969.
[4] L. B. Lucy, “A numerical approach to the testing of fission hypothesis,”
Astronomical Journal, vol. 8, pp. 1013-1024, 1977.
[5] B. Nayroles, G. Touzot, and P. Villon, “Generalizing the finite element
method: Diffuse approximation and diffuse elements,” Computational
Mechanics, vol. 10, no. 5, pp. 307-318, 1992.
[6] T. Belytschko, Y. Y. Lu, and L. Gu, “ELEMENT-FREE GALERKIN
METHODS,” International Journal for Numerical Methods in
Engineering, vol. 37, no. 2, pp. 229-256, 1994.
[7] W. K. Liu, S. Jun, S. F. Li, J. Adee, and T. Belytschko, “Reproducing
Kernel Particle Methods for Structural Dynamics,” International
Journal for Numerical Methods in Engineering, vol. 38, no. 10, pp.
1655-1679, May 30, 1995.
[8] J. M. Melenk, and I. Babuska, “The partition of unity finite element
method: Basic theory and applications,” Computer Methods in Applied
Mechanics and Engineering, vol. 139, no. 1-4, pp. 289-314, Dec 15,
1996.
[9] E. Onate, S. Idelsohn, O. C. Zienkiewicz, R. L. Taylor, and C. Sacco, “A
stabilized finite point method for analysis of fluid mechanics problems,”
Computer Methods in Applied Mechanics and Engineering, vol. 139, no.
1-4, pp. 315-346, Dec 15, 1996.
[10] C. Liu, X. Zheng, and C. Sung, “Preconditioned multigrid methods for
unsteady incompressible flows,” Journal of Computational Physics, vol.
139, no. 1, pp. 35-57, 1998.
[11] H. Ding, C. Shu, K. S. Yeo, and D. Xu, “Simulation of incompressible
viscous flows past a circular cylinder by hybrid FD scheme and
meshless least square-based finite difference method,” Computer
Methods in Applied Mechanics and Engineering, vol. 193, no. 9-11, pp.
727-744, 2004.
[12] Y. Sanyasiraju, and G. Chandhini, “Local radial basis function based
gridfree scheme for unsteady incompressible viscous flows,” Journal of
Computational Physics, vol. 227, no. 20, pp. 8922-8948, Oct, 2008.
[13] J. P. Morris, P. J. Fox, and Y. Zhu, “Modeling low Reynolds number
incompressible flows using SPH,” Journal of Computational Physics,
vol. 136, no. 1, pp. 214-226, Sep 1, 1997.
[14] J. J. Monaghan, “Simulating Free-Surface Flows with Sph,” Journal of
Computational Physics, vol. 110, no. 2, pp. 399-406, Feb, 1994.
[15] M. B. Liu, G. R. Liu, and K. Y. Lam, “Constructing smoothing functions
in smoothed particle hydrodynamics with applications,” Journal of
Computational and Applied Mathematics, vol. 155, no. 2, pp. 263-284,
Jun 15, 2003.
[16] M. Liu, G. Liu, Z. Zong, and K. Lam, "Numerical simulation of
incompressible flows by SPH."
[17] C. Shu, H. Ding., and K. S. Yeo., “Computation of Incompressible
Navier-Stokes Equations by Local RBF-based Differential Quadrature
Method,” Computer Modeling in Engineering and Sciences, vol. 7, no.
2, pp. 195-206, 2005.
[18] C. Shu, H. Ding, and K. S. Yeo, “Local radial basis function-based
differential quadrature method and its application to solve twodimensional
incompressible Navier-Stokes equations,” Computer
Methods in Applied Mechanics and Engineering, vol. 192, no. 7-8, pp.
941-954, 2003.
[19] C. G. Koh, M. Gao, and C. Luo, “A new particle method for simulation
of incompressible free surface flow problems,” International Journal for
Numerical Methods in Engineering, vol. 89, no. 12, pp. 1582-1604, Mar
23, 2012.
[20] S.-y. Tuann, and M. D. Olson, “Numerical studies of the flow around a
circular cylinder by a finite element method,” Computers & Fluids, vol.
6, no. 4, pp. 219-240, 12//, 1978.
[21] C. S. Chew, K. S. Yeo, and C. Shu, “A generalized finite-difference
(GFD) ALE scheme for incompressible flows around moving solid
bodies on hybrid meshfree–Cartesian grids,” Journal of Computational
Physics, vol. 218, no. 2, pp. 510-548, 11/1/, 2006.
[22] A. Belov, L. Martinelli, and A. Jameson, “A new implicit algorithm with
multigrid for unsteady incompressible flow calculations,” AIAA paper,
vol. 95, pp. 0049, 1995.
[23] X. He, and G. Doolen, “Lattice Boltzmann method on curvilinear
coordinates system: flow around a circular cylinder,” Journal of
Computational Physics, vol. 134, no. 2, pp. 306-315, 1997.
[24] R. Mei, and W. Shyy, “On the finite difference-based lattice Boltzmann
method in curvilinear coordinates,” Journal of Computational Physics,
vol. 143, no. 2, pp. 426-448, 1998.
[25] E. J. Kansa, “Multiquadrics - a Scattered Data Approximation Scheme
with Applications to Computational Fluid-Dynamics .2. Solutions to
Parabolic, Hyperbolic and Elliptic Partial-Differential Equations,”
Computers & Mathematics with Applications, vol. 19, no. 8-9, pp. 147-
161, 1990.
[26] J. G. Wang, and G. R. Liu, “On the optimal shape parameters of radial
basis functions used for 2-D meshless methods,” Computer Methods in
Applied Mechanics and Engineering, vol. 191, no. 23-24, pp. 2611-
2630, 2002.
[27] P. P. Chinchapatnam, K. Djidjeli, and P. B. Nair, “Radial basis function
meshless method for the steady incompressible Navier–Stokes
equations,” International Journal of Computer Mathematics, vol. 84, no.
10, pp. 1509-1521, 2007.
[28] Z. Guo, B. Shi, and N. Wang, “Lattice BGK model for incompressible
Navier–Stokes equation,” Journal of Computational Physics, vol. 165,
no. 1, pp. 288-306, 2000.
[29] A. I. Tolstykh, and D. A. Shirobokov, “On using radial basis functions
in a "finite difference mode" with applications to elasticity problems,”
Computational Mechanics, vol. 33, no. 1, pp. 68-79, Dec, 2003.
[30] G. B. Wright, and B. Fornberg, “Scattered node compact finite
difference-type formulas generated from radial basis functions,” Journal
of Computational Physics, vol. 212, no. 1, pp. 99-123, Feb 10, 2006.
[31] A. R. Firoozjaee, and M. H. Afshar, “Steady-state solution of
incompressible Navier–Stokes equations using discrete least-squares
meshless method,” International Journal for Numerical Methods in
Fluids, vol. 67, no. 3, pp. 369-382, 2011.
[32] P. P. Chinchapatnam, K. Djidjeli, P. B. Nair, and M. Tan, “A compact
RBF-FD based meshless method for the incompressible Navier-Stokes
equations,” Proceedings of the Institution of Mechanical Engineers Part
M-Journal of Engineering for the Maritime Environment, vol. 223, no.
M3, pp. 275-290, Aug, 2009.
[33] C. Perng, and R. Street, “A coupled multigrid‐domain‐splitting
technique for simulating incompressible flows in geometrically complex
domains,” International journal for numerical methods in fluids, vol. 13,
no. 3, pp. 269-286, 1991.
[34] M. Hinatsu, and J. Ferziger, “Numerical computation of unsteady
incompressible flow in complex geometry using a composite multigrid
technique,” International Journal for Numerical Methods in Fluids, vol.
13, no. 8, pp. 971-997, 1991.
[35] P. Chow, and C. Addison, “Putting domain decomposition at the heart of
a mesh‐based simulation process,” International journal for numerical
methods in fluids, vol. 40, no. 12, pp. 1471-1484, 2002.
[36] D. Calhoun, “A Cartesian grid method for solving the two-dimensional
streamfunction-vorticity equations in irregular regions,” Journal of
Computational Physics, vol. 176, no. 2, pp. 231-275, 2002.
[37] R. B. Pember, J. B. Bell, P. Colella, W. Y. Curtchfield, and M. L.
Welcome, “An adaptive Cartesian grid method for unsteady
compressible flow in irregular regions,” Journal of computational
Physics, vol. 120, no. 2, pp. 278-304, 1995.
[38] J. Falcovitz, G. Alfandary, and G. Hanoch, “A two-dimensional
conservation laws scheme for compressible flows with moving
boundaries,” Journal of Computational Physics, vol. 138, no. 1, pp. 83-
102, 1997.
[39] A. Gilmanov, F. Sotiropoulos, and E. Balaras, “A general reconstruction
algorithm for simulating flows with complex 3D immersed boundaries
on Cartesian grids,” Journal of Computational Physics, vol. 191, no. 2,
pp. 660-669, 2003.
[40] R. Glowinski, T.-W. Pan, and J. Periaux, “A fictitious domain method
for external incompressible viscous flow modeled by Navier-Stokes
equations,” Computer Methods in Applied Mechanics and Engineering,
vol. 112, no. 1, pp. 133-148, 1994.
[41] A. Javed, K. Djidjeli, and J. Xing, Tang, “Shape adaptive RBF-FD
Implicit Scheme for Incompressible Viscous Navier-Strokes Equations,”
Journal of Computer and Fluid, vol. submitted for Publication, 2013.
[42] J. Kim, and P. Moin, “Application of a Fractional-Step Method to
Incompressible Navier-Stokes Equations,” Journal of Computational
Physics, vol. 59, no. 2, pp. 308-323, June, 1985, 1985.
[43] B. Fornberg, “A Numerical Study of Steady Viscous-Flow Past a
Circular-Cylinder,” Journal of Fluid Mechanics, vol. 98, no. Jun, pp.
819-855, 1980. [44] D. Kim, and H. Choi, “A second-order time-accurate finite volume
method for unsteady incompressible flow on hybrid unstructured grids,”
Journal of Computational Physics, vol. 162, no. 2, pp. 411-428, 2000.
[45] Y. Zang, R. L. Street, and J. R. Koseff, “A non-staggered grid, fractional
step method for time-dependent incompressible Navier-Stokes equations
in curvilinear coordinates,” Journal of Computational Physics, vol. 114,
no. 1, pp. 18-33, 1994.
[1] S. Dennis, and G.-Z. Chang, “Numerical solutions for steady flow past a
circular cylinder at Reynolds numbers up to 100,” J. Fluid Mech, vol.
42, no. 3, pp. 471-489, 1970.
[2] M. Braza, P. Chassaing, and H. H. Minh, “Numerical Study and
Physical Analysis of the Pressure and Velocity-Fields in the near Wake of a Circular-Cylinder,” Journal of Fluid Mechanics, vol. 165, pp. 79-
130, Apr, 1986.
[3] H. Takami, and H. B. Keller, “Steady Two‐Dimensional Viscous Flow
of an Incompressible Fluid past a Circular Cylinder,” Physics of Fluids,
vol. 12, no. 12, pp. II-51-II-56, 1969.
[4] L. B. Lucy, “A numerical approach to the testing of fission hypothesis,”
Astronomical Journal, vol. 8, pp. 1013-1024, 1977.
[5] B. Nayroles, G. Touzot, and P. Villon, “Generalizing the finite element
method: Diffuse approximation and diffuse elements,” Computational
Mechanics, vol. 10, no. 5, pp. 307-318, 1992.
[6] T. Belytschko, Y. Y. Lu, and L. Gu, “ELEMENT-FREE GALERKIN
METHODS,” International Journal for Numerical Methods in
Engineering, vol. 37, no. 2, pp. 229-256, 1994.
[7] W. K. Liu, S. Jun, S. F. Li, J. Adee, and T. Belytschko, “Reproducing
Kernel Particle Methods for Structural Dynamics,” International
Journal for Numerical Methods in Engineering, vol. 38, no. 10, pp.
1655-1679, May 30, 1995.
[8] J. M. Melenk, and I. Babuska, “The partition of unity finite element
method: Basic theory and applications,” Computer Methods in Applied
Mechanics and Engineering, vol. 139, no. 1-4, pp. 289-314, Dec 15,
1996.
[9] E. Onate, S. Idelsohn, O. C. Zienkiewicz, R. L. Taylor, and C. Sacco, “A
stabilized finite point method for analysis of fluid mechanics problems,”
Computer Methods in Applied Mechanics and Engineering, vol. 139, no.
1-4, pp. 315-346, Dec 15, 1996.
[10] C. Liu, X. Zheng, and C. Sung, “Preconditioned multigrid methods for
unsteady incompressible flows,” Journal of Computational Physics, vol.
139, no. 1, pp. 35-57, 1998.
[11] H. Ding, C. Shu, K. S. Yeo, and D. Xu, “Simulation of incompressible
viscous flows past a circular cylinder by hybrid FD scheme and
meshless least square-based finite difference method,” Computer
Methods in Applied Mechanics and Engineering, vol. 193, no. 9-11, pp.
727-744, 2004.
[12] Y. Sanyasiraju, and G. Chandhini, “Local radial basis function based
gridfree scheme for unsteady incompressible viscous flows,” Journal of
Computational Physics, vol. 227, no. 20, pp. 8922-8948, Oct, 2008.
[13] J. P. Morris, P. J. Fox, and Y. Zhu, “Modeling low Reynolds number
incompressible flows using SPH,” Journal of Computational Physics,
vol. 136, no. 1, pp. 214-226, Sep 1, 1997.
[14] J. J. Monaghan, “Simulating Free-Surface Flows with Sph,” Journal of
Computational Physics, vol. 110, no. 2, pp. 399-406, Feb, 1994.
[15] M. B. Liu, G. R. Liu, and K. Y. Lam, “Constructing smoothing functions
in smoothed particle hydrodynamics with applications,” Journal of
Computational and Applied Mathematics, vol. 155, no. 2, pp. 263-284,
Jun 15, 2003.
[16] M. Liu, G. Liu, Z. Zong, and K. Lam, "Numerical simulation of
incompressible flows by SPH."
[17] C. Shu, H. Ding., and K. S. Yeo., “Computation of Incompressible
Navier-Stokes Equations by Local RBF-based Differential Quadrature
Method,” Computer Modeling in Engineering and Sciences, vol. 7, no.
2, pp. 195-206, 2005.
[18] C. Shu, H. Ding, and K. S. Yeo, “Local radial basis function-based
differential quadrature method and its application to solve twodimensional
incompressible Navier-Stokes equations,” Computer
Methods in Applied Mechanics and Engineering, vol. 192, no. 7-8, pp.
941-954, 2003.
[19] C. G. Koh, M. Gao, and C. Luo, “A new particle method for simulation
of incompressible free surface flow problems,” International Journal for
Numerical Methods in Engineering, vol. 89, no. 12, pp. 1582-1604, Mar
23, 2012.
[20] S.-y. Tuann, and M. D. Olson, “Numerical studies of the flow around a
circular cylinder by a finite element method,” Computers & Fluids, vol.
6, no. 4, pp. 219-240, 12//, 1978.
[21] C. S. Chew, K. S. Yeo, and C. Shu, “A generalized finite-difference
(GFD) ALE scheme for incompressible flows around moving solid
bodies on hybrid meshfree–Cartesian grids,” Journal of Computational
Physics, vol. 218, no. 2, pp. 510-548, 11/1/, 2006.
[22] A. Belov, L. Martinelli, and A. Jameson, “A new implicit algorithm with
multigrid for unsteady incompressible flow calculations,” AIAA paper,
vol. 95, pp. 0049, 1995.
[23] X. He, and G. Doolen, “Lattice Boltzmann method on curvilinear
coordinates system: flow around a circular cylinder,” Journal of
Computational Physics, vol. 134, no. 2, pp. 306-315, 1997.
[24] R. Mei, and W. Shyy, “On the finite difference-based lattice Boltzmann
method in curvilinear coordinates,” Journal of Computational Physics,
vol. 143, no. 2, pp. 426-448, 1998.
[25] E. J. Kansa, “Multiquadrics - a Scattered Data Approximation Scheme
with Applications to Computational Fluid-Dynamics .2. Solutions to
Parabolic, Hyperbolic and Elliptic Partial-Differential Equations,”
Computers & Mathematics with Applications, vol. 19, no. 8-9, pp. 147-
161, 1990.
[26] J. G. Wang, and G. R. Liu, “On the optimal shape parameters of radial
basis functions used for 2-D meshless methods,” Computer Methods in
Applied Mechanics and Engineering, vol. 191, no. 23-24, pp. 2611-
2630, 2002.
[27] P. P. Chinchapatnam, K. Djidjeli, and P. B. Nair, “Radial basis function
meshless method for the steady incompressible Navier–Stokes
equations,” International Journal of Computer Mathematics, vol. 84, no.
10, pp. 1509-1521, 2007.
[28] Z. Guo, B. Shi, and N. Wang, “Lattice BGK model for incompressible
Navier–Stokes equation,” Journal of Computational Physics, vol. 165,
no. 1, pp. 288-306, 2000.
[29] A. I. Tolstykh, and D. A. Shirobokov, “On using radial basis functions
in a "finite difference mode" with applications to elasticity problems,”
Computational Mechanics, vol. 33, no. 1, pp. 68-79, Dec, 2003.
[30] G. B. Wright, and B. Fornberg, “Scattered node compact finite
difference-type formulas generated from radial basis functions,” Journal
of Computational Physics, vol. 212, no. 1, pp. 99-123, Feb 10, 2006.
[31] A. R. Firoozjaee, and M. H. Afshar, “Steady-state solution of
incompressible Navier–Stokes equations using discrete least-squares
meshless method,” International Journal for Numerical Methods in
Fluids, vol. 67, no. 3, pp. 369-382, 2011.
[32] P. P. Chinchapatnam, K. Djidjeli, P. B. Nair, and M. Tan, “A compact
RBF-FD based meshless method for the incompressible Navier-Stokes
equations,” Proceedings of the Institution of Mechanical Engineers Part
M-Journal of Engineering for the Maritime Environment, vol. 223, no.
M3, pp. 275-290, Aug, 2009.
[33] C. Perng, and R. Street, “A coupled multigrid‐domain‐splitting
technique for simulating incompressible flows in geometrically complex
domains,” International journal for numerical methods in fluids, vol. 13,
no. 3, pp. 269-286, 1991.
[34] M. Hinatsu, and J. Ferziger, “Numerical computation of unsteady
incompressible flow in complex geometry using a composite multigrid
technique,” International Journal for Numerical Methods in Fluids, vol.
13, no. 8, pp. 971-997, 1991.
[35] P. Chow, and C. Addison, “Putting domain decomposition at the heart of
a mesh‐based simulation process,” International journal for numerical
methods in fluids, vol. 40, no. 12, pp. 1471-1484, 2002.
[36] D. Calhoun, “A Cartesian grid method for solving the two-dimensional
streamfunction-vorticity equations in irregular regions,” Journal of
Computational Physics, vol. 176, no. 2, pp. 231-275, 2002.
[37] R. B. Pember, J. B. Bell, P. Colella, W. Y. Curtchfield, and M. L.
Welcome, “An adaptive Cartesian grid method for unsteady
compressible flow in irregular regions,” Journal of computational
Physics, vol. 120, no. 2, pp. 278-304, 1995.
[38] J. Falcovitz, G. Alfandary, and G. Hanoch, “A two-dimensional
conservation laws scheme for compressible flows with moving
boundaries,” Journal of Computational Physics, vol. 138, no. 1, pp. 83-
102, 1997.
[39] A. Gilmanov, F. Sotiropoulos, and E. Balaras, “A general reconstruction
algorithm for simulating flows with complex 3D immersed boundaries
on Cartesian grids,” Journal of Computational Physics, vol. 191, no. 2,
pp. 660-669, 2003.
[40] R. Glowinski, T.-W. Pan, and J. Periaux, “A fictitious domain method
for external incompressible viscous flow modeled by Navier-Stokes
equations,” Computer Methods in Applied Mechanics and Engineering,
vol. 112, no. 1, pp. 133-148, 1994.
[41] A. Javed, K. Djidjeli, and J. Xing, Tang, “Shape adaptive RBF-FD
Implicit Scheme for Incompressible Viscous Navier-Strokes Equations,”
Journal of Computer and Fluid, vol. submitted for Publication, 2013.
[42] J. Kim, and P. Moin, “Application of a Fractional-Step Method to
Incompressible Navier-Stokes Equations,” Journal of Computational
Physics, vol. 59, no. 2, pp. 308-323, June, 1985, 1985.
[43] B. Fornberg, “A Numerical Study of Steady Viscous-Flow Past a
Circular-Cylinder,” Journal of Fluid Mechanics, vol. 98, no. Jun, pp.
819-855, 1980. [44] D. Kim, and H. Choi, “A second-order time-accurate finite volume
method for unsteady incompressible flow on hybrid unstructured grids,”
Journal of Computational Physics, vol. 162, no. 2, pp. 411-428, 2000.
[45] Y. Zang, R. L. Street, and J. R. Koseff, “A non-staggered grid, fractional
step method for time-dependent incompressible Navier-Stokes equations
in curvilinear coordinates,” Journal of Computational Physics, vol. 114,
no. 1, pp. 18-33, 1994.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:65242", author = "A. Javed and K. Djidjeli and J. T. Xing and S. J. Cox", title = "A Hybrid Mesh Free Local RBF- Cartesian FD Scheme for Incompressible Flow around Solid Bodies", abstract = "A method for simulating flow around the solid bodies has been presented using hybrid meshfree and mesh-based schemes. The presented scheme optimizes the computational efficiency by combining the advantages of both meshfree and mesh-based methods. In this approach, a cloud of meshfree nodes has been used in the domain around the solid body. These meshfree nodes have the ability to efficiently adapt to complex geometrical shapes. In the rest of the domain, conventional Cartesian grid has been used beyond the meshfree cloud. Complex geometrical shapes can therefore be dealt efficiently by using meshfree nodal cloud and computational efficiency is maintained through the use of conventional mesh-based scheme on Cartesian grid in the larger part of the domain. Spatial discretization of meshfree nodes has been achieved through local radial basis functions in finite difference mode (RBF-FD). Conventional finite difference scheme has been used in the Cartesian ‘meshed’ domain. Accuracy tests of the hybrid scheme have been conducted to establish the order of accuracy. Numerical tests have been performed by simulating two dimensional steady and unsteady incompressible flows around cylindrical object. Steady flow cases have been run at Reynolds numbers of 10, 20 and 40 and unsteady flow problems have been studied at Reynolds numbers of 100 and 200. Flow Parameters including lift, drag, vortex shedding, and vorticity contours are calculated. Numerical results have been found to be in good agreement with computational and experimental results available in the literature.
", keywords = "CFD, Meshfree particle methods, Hybrid grid, Incompressible Navier Strokes equations, RBF-FD.", volume = "7", number = "10", pages = "1494-10", }
